Question

Two numbers are drawn, each in the range {0,1}. What is the probability that their sum is less than 1 while their product is greater than 1/5?

Answer 1

This is quite a straightforward geometric probability problem. Here the approach is to assume the given variables along the coordinate axes, and find the area of the required feasible region. The sample space is the defined range of given variables.

Hence, we have sample space as the unit square in the first quadrant, with one vertex at origin. That is,

The area of the feasible region is  

The required region is the intersection of the inequations

This region looks like

The two curves intersect at two points between which their common area is bounded, given by

The area of required region is

Inserting the values, we get

So the required probability is

There is about 3.1% probability that the numbers satisfy given conditions